In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications.
Geometrically, there are two ways to describe trigonometric functions:
Polar Angle
$\begin{array}{l}
x=\cos\theta\\
y=\sin\theta\\
~\\
{\small\textrm{Measure }} \theta {\small\textrm{ in radians:}}\\
\theta =\frac{{\small\textrm{arc length}}}{{\small\textrm{radius}}}\\
~\\
{\small\textrm{For example,}}\quad 180^{\circ}=\displaystyle\frac{\pi r}{r}=\pi
{\small\textrm{ radians}}\\
~\\
{\small\textrm{Radians}}=\displaystyle\frac{{\small\textrm{degrees}}}{180}\cdot \pi
\end{array}$
Right Angle
$\begin{array}{l}
\sin\theta &=& \displaystyle \frac{{\small\textrm{opposite}}}{{\small\textrm{hypotenuse}}} &=& \frac{y}{r}\\~\\
\cos\theta &=& \displaystyle \frac{{\small\textrm{adjacent}}}{{\small\textrm{hypotenuse}}} &=& \frac{x}{r}\\~\\
\tan\theta &=& \displaystyle \frac{{\small\textrm{opposite}}}{{\small\textrm{adjacent}}} &=& \frac{y}{x}\\~\\
\csc\theta &=& \displaystyle \frac{1}{\sin\theta} &=& \frac{r}{y}\\~\\
\sec\theta &=& \displaystyle \frac{1}{\cos\theta} &=& \frac{r}{x}\\~\\
\cot\theta &=& \displaystyle \frac{1}{\tan\theta} &=& \frac{x}{y}
\end{array}$
Evaluating Trigonometric Functions
| $0 {\small\textrm{ rad}} $ | $\pi/6 {\small\textrm{ rad}} $ | $\pi/4 {\small\textrm{ rad}} $ | $\pi/3 {\small\textrm{ rad}} $ | $\pi/2 {\small\textrm{ rad}}$ | |
| $0^{\circ} $ | $30^{\circ} $ | $45^{\circ} $ | $60^{\circ} $ | $90^{\circ}$ | |
| $\sin\theta $ | $0 $ | $1/2 $ | $\sqrt{2}/2 $ | $\sqrt{3}/2 $ | $1$ |
| $\cos\theta $ | $1 $ | $\sqrt{3}/2 $ | $\sqrt{2}/2 $ | $1/2 $ | $0$ |
| $\tan\theta $ | $0 $ | $\sqrt{3}/3 $ | $1 $ | $\sqrt{3} $ | ${\small\textrm{undefined}}$ |
Trigonometric Identities
We list here some of the most commonly used identities:
$\begin{array}{lcr} \textbf{1. }\cos^2\theta+\sin^2\theta=1 \\ \textbf{2. }\cos^2\theta =\displaystyle\frac{1}{2}[1+\cos(2\theta)] \\ \textbf{3. } \sin^2\theta=\displaystyle\frac{1}{2}[1-\cos(2\theta)]\\ \textbf{4. } \sin(2\theta)=2\sin\theta\cos\theta \\ \textbf{5. }\cos(2\theta)=\cos^2\theta-\sin^2\theta \end{array}$
$\begin{array}{lcr}
\textbf{6. } \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\\
\textbf{7. } \cos(\alpha +\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\
~\\
\textbf{8. } C_1\cos(\omega x)+C_2\sin(\omega x)=A\sin(\omega x+\phi)\\
\qquad{\small\textrm{where }} A=\sqrt{C_1^2+C_2^2},\quad \phi=\arctan (C_1/C_2)
\end{array}$
Logarithmic and Exponential Functions
Logarithmic and exponential functions are inverses of each other: \begin{eqnarray*} y=\log_b x & \quad{\small\textrm{if and only if}} & x=b^y\\ y=\ln x & {\small\textrm{ if and only if }} & x=e^y. \end{eqnarray*} In words, $\displaystyle \log_b x$ is the exponent you put on base $b$ to get $x$. Thus, \[log_b b^x=x \qquad {\small\textrm{and}} \qquad b^{\log_b x}=x.\]
More Properties of Logarithmic and Exponential Functions
Notice the relationship between each pair of identities: \[\begin{array}{ccc@{\qquad}ccc} \log_b 1=0 & \longleftrightarrow & b^0=1 & \log_b ac=\log_b a+\log_b c & \longleftrightarrow & b^mb^n=b^{m+n}\\ \log_b b=1 & \longleftrightarrow & b^1=b & \log_b \displaystyle\frac{a}{c}=\log_b a-\log_b c & \longleftrightarrow & \displaystyle\frac{b^m}{b^n}=b^{m-n}\\ \log_b \displaystyle\frac{1}{c}=-\log_b c & \longleftrightarrow & b^{-m}=\displaystyle\frac{1}{b^m} & \log_b a^r=r\log_b a & \longleftrightarrow & (b^m)^n=b^{mn} . \end{array}\]
Graphs of Logarithmic and Exponential Functions
Limits of Logarithmic and Exponential Functions
- $\displaystyle \lim_{x\to\infty} \frac{\ln x}{x}=0\quad$ $\ln x$ grows more slowly than $x$.
- $\displaystyle \lim_{x\to\infty} \frac{e^x}{x^n}=\infty$ for all positive integers $n\quad$, $\displaystyle e^x$ grows faster than $x^n$.
- For $|x|\ll 1$, $\displaystyle\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n=e^x$.